UNIVERSITI PUTRA MALAYSIA

OUTLIER DETECTIONS AND ROBUST ESTIMATION METHODS FOR

NONLINEAR REGRESSION MODEL HAVING AUTOCORRELATED AND

HETEROSCEDASTIC ERRORS

HOSSEIN RIAZOSHAMS

IPM 2010 13

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OUTLIER DETECTIONS AND ROBUST ESTIMATION METHODS FOR NONLINEAR REGRESSION MODEL HAVING AUTOCORRELATED AND

HETEROSCEDASTIC ERRORS

By

HOSSEIN RIAZOSHAMS

Thesis Submitted to the School of Graduate Studies, Universiti Putra Malaysia, in Fulfilment of the Requirements for the Degree of Doctor of Philosophy

November 2010

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Dedicated to:

My father

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Abstract of the thesis presented to the Senate of Universiti Putra Malaysia in fulfilments for the degree of Doctor of Philosophy

OUTLIER DETECTIONS AND ROBUST ESTIMATION METHODS FOR

NONLINEAR REGRESSION MODEL HAVING AUTOCORRELATED AND HETEROSCEDASTIC ERRORS

By

Hossein Riazoshams

November 2010

Chairman: Associate Professor Habshah Midi, PhD

Faculty: INSPEM

The ordinary Nonlinear Least Squares (NLLS) and the Maximum Likelihood

Estimator (MLE) techniques are often used to estimate the parameters of nonlinear

models. Unfortunately, many researchers are not aware of the consequences of using

such estimators when outliers are present in the data. The problems get more

complex when the assumption of constant error variances or homoscedasticity is

violated. To remedy these two problems simultaneously, we proposed a Robust

Multistage Estimator (RME).

The heterogeneouity of error variances is considered when the variances of residuals

follows a parametric functional form of the predictors. Both Nonlinear model

function parameters and variance model parameters must be robustified. We have

incorporated the MM, the generalized MM and the robustified Chi-Squares Pseudo

Likelihood function in the formulation of the RME. The results of the study reveal

that the RME is more efficient than the existing methods.

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The thesis also addresses the problems when the assumptions of the independent

error terms are not met. We proposed a new Robust Two Stage (RTS) estimator in

this regard. The proposed method is developed by incorporating the generalized MM

estimator in the classical two stage estimator. The performance of the RTS is more

efficient than other existing methods revealed by having the highest robustness

measures.

We also proposed two outlier identification measures in nonlinear regression. The

Tangent leverage, the NLLS, the M and the MM estimators are incorporated in the

formulation of the first outlier identification measures. The formulation of the second

measure is based on the differences between the derived robust Jacobian Leverage

and Tangent leverage. Both proposed measures are very successful to identify the

correct outliers.

Finally, we proposed statistics practitioners to use the formal modeling algorithms to

get better inferences. We also suggest them to employ appropriate robust methods for

further analysis once a correct model has been chosen. The results of the study based

on real data signify that the robust estimator is more efficient indicated by lower

values of standard errors when compared to the classical estimator.

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Abstrak tesis dikemukakan kepada Senat Uiversiti Putra Malaysia sebagai memenuhi keperluan untuk Ijazah Doktor Falsafah

PENGESANAN DATA TERPENCIL DAN KAEDAH PENGANGGARAN

TEGUH BAGI MODEL TAK LINEAR YANG MEMPUNYAI RALAT BERAUTOKORELASI DAN BERHETEROSKEDASTIK.

Oleh

HOSSEIN RIAZOSHAMS

November 2010

Pengerusi: Profesor Madya Habshah Midi, PhD

Faculty: INSPEM

Teknik Kuasa Dua Terkecil Tak Linear (NLLS) dan Penganggar Kebolehjadian

Maksimum (MLE) kerap digunakan untuk menganggar parameter model tak linear.

Malangnya, kebanyakan penyelidik kurang peka akan akibat penggunaan teknik

tersebut jika terdapat titik terpancil pada data. Masalah ini semakin kompleks apabila

andaian varians beracat tetap atau berhomoskedastisiti tidak dipatuhi. Untuk

menyelesaikan dua permasalahan ini secara serentak, kami mencadangkan

Penganggar Multiperingkat Teguh (RME).

Varians ralat yang berbeza dipertimbangkan apabila varians reja mematuhi bentuk

fungsi parametrik beberapa pembolehubuh tak bersandaran. Kedua-dua fungsi

parameter bagi model tak linear dan model varians bagi parameter mesti diteguhkan.

Kami menggabungkan penganggar M, penganggar MM teritlak dan juga fungsi

teguh kebolehjadian Pseudo Khi Kuasa Dua ke dalam pembentukan PME Dapatan

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daripada kajian ini menunjukkan RME lebih efisien berbanding kaedah yang sedia

ada.

Kajian ini juga menyebut permasaalahan apabila andaian ralat bebas tidak dipenuhi.

Oleh itu kami mencadangkan Penganggar Dua Langkah Teguh (RTS) untuk

permasalahan ini. Kaedah cadangan ini dibina dengan menggabungkan penganggar

MM teritlak kedalam penganggar dua langkah klasik. Keupayaan RTS didapati lebih

efisien berbanding kaedah terdahulu, kerana ianya mempunyai ukuran keteguhan

yang tinggi.

Kami juga mencadangkan dua pengukuran pegesanan data terpencil dalam regresi

tak linear. Tuasan Tangen digabungkan bersama penganggar NLLS, penganggar M

dan penganggar MM ke dalam formulasi pengukuran pengesanan titik terpencik

yang pertama. Formulasi kedua pula berasaskan pada perbezaan diantara tuasan

teguh Jacobian yang diterbitkan dan tuasan Tangen. Kedua-dua pengukuran

cadangan ini sangat berkesan dalam menentukan titik terpencil yang sebenar.

Akhir sekali, kami mencadangkan kepada pengamal statistik untuk menggunakan

pemodelan algoritma yang formal untuk mendapatkan inferens yang lebih baik.

Kami juga mencadangkan mereka menggunakan kaedah teguh yang sesuai untuk

melakukan analisis lanjutan setelah model yang betul dipilih. Hasil keputusan kajian

berdasarkan pada data sebenar, menunjukkan bahawa penganggar teguh adalah lebih

efisien, ditunjukkan dari nilai ralat piawai yang rendah berbanding nilai penganggar

klasik.

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ACKNOWLEDGEMENTS

With the merciful of God that gave me strength to understand science.

I would like to express my deep gratitude and warmest thanks to my supervisor,

Associate Professor Dr. Habshah Midi for the great academic help and her

continuous guidance and encouraging, during the course of my study.

In addition I would like to thank my supervisory comity members, Dr. Bakri Adam,

Dr. Ibragimof Gafourjon, for their valuable discussions and help.

Special thanks to Professor Olimjon Sh. Sharipov. Statistics Professor from Institute

of Mathematics and Information Technology Uzbek Academy of Science, Tashkent,

Uzbekistan, for his useful remarks and being the co-author of our best presented

paper.

I gratefully acknowledge the financial support from Universiti Putra Malaysia,

during my study, special thanks to Associate Professor Dr. Habshah Midi, and Dr.

Bakri Adam, also for their financial support.

I would like to thanks, my Lecturers and Colleagues, and friends, Dr. Hossein

Marzban, Economic Department, Shiraz University, Dr Ahmad Parsian Department

of statistics, Shiraz University, Prof Javad Behboodian, Shiraz University, Benjamat

Hanchana, velo suthar, ali edalati, Sarkhosh Sedighic for their kindly encouraging

me to continue to PHD study. And special thanks to my sister Fariba Riazoshams for

her helpful advises during my PHD study. And special appreciation to Amir ali,

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arsalan sedaghat pisheh, Mohsen Solhdoost, Mohammed Ashyraf, for editing my

thesis.

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I certify that an Examination Committee met on 3rt Nov 2010 to conduct the final examination of Hossein Riazoshams on his Doctor of Philosophy thesis entitled “OUTLIER DETECTIONS AND ROBUST ESTIMATION METHODS FOR NONLINEAR REGRESSION MODEL HAVING AUTOCORRELATED AND HETEROSCEDASTIC ERRORS” in accordance with Universiti Putra Malaysia (Higher Degree) Act 1980 and Universiti Putra Malaysia (Higher Degree) Regulations 1981. The Committee recommends that the student be awarded the Doctor of Philosophy. Members of the Examination Committee were as follows:

Dr Isa bin Duad, PhD Associate Professor. Faculty of Science Universiti Putra Malaysia (Chairman) Kassim bin Haron, PhD Profesor Madya Dr. Faculty of Science Universiti Putra Malaysia (Internal Examiner) Mohd Rizam Abu Bakar, PhD Profesor Madya Dr. Faculty of Science Universiti Putra Malaysia (Internal Examiner) Zuhair A. Al-Hemyari, PhD Profesor College of Art and Science Universiti of Nizwa. Sultanate of Oman. (External Examiner) _______________________________ HASANAH MOHD. GHAZALI, PhD Professor and Deputy Dean School of Graduate Studies Universiti Putra Malaysia Date:

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This thesis was submitted to the Senate of University Putra Malaysia and has been accepted as fulfilment of requirement for the degree of Doctor of Philosophy. The members of Supervisory Committee were as follows.

Habshah Midi, PhD Associate Professor Institute for Mathematical Research Universiti Putra Malaysia (Chairman) Bakri Adam, PhD Assistant Proffesor Institute for Mathematical Research Universiti Putra Malaysia (Member) Ibragimov Garfounjan, PhD Associate Professor Faculty of Science Universiti Putra Malaysia (Member)

_______________________________ HASANAH MOHD GHAZALI, PhD Professor and Dean School of Graduate Studies Universiti Putra Malaysia Date:

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DECLARATION

I declare that the thesis is my original work except for quotations and citations, which have been duly acknowledged. I also declare that it has not been previously, and is not concurrently submitted for any other degree at Universiti Putra Malaysia or institution.

____________________________

HOSSEIN RIAZOSHAMS

Date:

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OPYRIGHT U

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Table of Contents

Page

ABSTRACT iii ABSTRAK v ACKNOWLEDGEMENTS vii APPROVAL ix DECLARATION xi LIST OF TABLES xv LIST OF FIGURES xvii ABREVIATIONS xx

1 INTRODUCTION 1 1.1 Background 1 1.2 Motivation of Study 2 1.3 Research Objectives 6 1.4 Significance of Study 7 1.5 Scope and Limitation of the Study 8 1.6 Outline of the Thesis 10

2 LITRATURE REVIEW 12 2.1 Introduction 12 2.2 Nonlinear Models 13 2.3 Robust Regression 14 2.4 Parameter Estimation 18 2.5 Outlier Detection 24 2.6 Heteroscedasticity of Variance 26 2.7 Autocorrelated Errors 28 2.8 Modeling Nonlinear Data 30

3 MATHEMATICAL BACKGROUND 32 3.1 Introduction 32 3.2 Nonlinear Model 32

3.2.1 The Ordinary Least Square (NLLS) 33 3.2.2 Modified Gauss-Newton (Constant Variance) 35 3.2.3 The Curvature Measure of Nonlinearity 37

3.3 The Generalized Least Squares Estimator 41 3.4 Robust MM-Estimator 42 3.5 Generalized M and MM-Estimators 44 3.6 Optimization 44

3.6.1 Iterative Methods 46 3.6.2 Convergence Criteria 48 3.6.3 Combined Algorithm 50 3.6.4 Robust M-estimator 50 3.6.5 The Generalized M-Estimator 52

3.7 Some Mathematical Notation 52

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4 OUTLIER DETECTION MEASURE BASED ON TANGENT LEVEREAGE AND NLLS, M AND MM ESTIMATOR 54 4.1 Introduction 54 4.2 Estimation Methods 55 4.3 The Outlier Detection Measures 56

4.3.1 Studentized and Deletion Studentized Residuals 58 4.3.2 Hadi’s Potential 59 4.3.3 Elliptic Norm (Cook Distance) 60 4.3.4 Difference in Fits 60 4.3.5 Atkinson’s Distance 61

4.4 Numerical Example 61 4.5 Simulation Study 63 4.6 Conclusion 68

5 A NEW DIRECT OUTLIER DETECTION MEASURE BASED ON THE DIFFERENCE BETWEEN THE ROBUST LEVERAGE AND TANGENT LEVERAGE 78 5.1 Introduction 78 5.2 Jacobian Leverage 79 5.3 The Generalized and Jacobian Leverage for M-Estimator 82 5.4 Outlier Detection Measures based on Jacobian Leverage, Robust

Jacobian Leverage and M & MM Estimators 86 5.5 Numerical Example To Assess the Performance of the New

Measures Based on Jacobian and Robust Jacobian Leverages 87 5.6 Simulation Study To Assess the Performance of the New

Measures Based on Jacobian and Robust Jacobian Leverages 88 5.7 Robust Jacobian Leverage and Local Influences 98 5.8 Numerical Example To Assess the Performance of DLev

Measure 103 5.9 Simulation Study To Assess the Performance of DLev Measure 104 5.10 Conclusion 111

6 ROBUST ESTIMATOR OF NONLINEAR REGRESSION WITH HETEROSCEDASTIC ERROR VARIANCES 112 6.1 Introduction 112 6.2 Model Parameter Estimation 116 6.3 Variance Modeling and Estimation 117 6.4 Robust Multistage Estimate 122 6.5 Numerical Optimization (RME) 130 6.6 Numerical Example 133 6.7 Monte Carlo study 137 6.8 Conclusion 138

7 THE PERFORMANCE OF ROBUST TWO STAGE ESTIMATOR IN NONLINEAR REGRESSION WITH AUTOCORRELATED ERROR 142 7.1 Introduction 142 7.2 Nonlinear Autocorrelated Model 144 7.3 The Classical Two Stage Estimator 146

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7.4 Robust Two Stage Estimator 147 7.5 Numerical Example 149 7.6 Monte Carlo Simulation 152 7.7 Conclusion 156

8 A NONLINEAR MODELING APPROACH WHEN THE DATA HAVING HETEROSCEDASTIC ERRORS 166 8.1 Introduction 166 8.2 Commonly Used Modeling Techniques. 167

8.2.1 Heteroscedastic Error 168 8.3 A Formal Technique for Model selection 171

8.3.1 Variance Modeling and Estimation 173 8.4 Applying the Commonly Used Model Selection Technique for

Chicken Growth Data 174 8.5 Applying Formal Model Selection Technique (Bunke (1995 a

and b)) to Chicken Growth Data 180 8.6 Further Analysis After The Model Selection Result 187 8.7 Conclusion 188

9 GENERAL SUMMARY CONCLUSIONS AND RECOMMENDATION FOR FUTURE RESEARCH 189 9.1 Introduction 189 9.2 Optimization 190 9.3 Contribution of the Study 190

9.3.1 Outlier Detection Measure Based on Tangent Leverage and NLLS, M and MM Estimator 190

9.3.2 A New Direct Outlier Detection Measure Based on the Difference Between the Robust Leverage and Tangent Leverage 191

9.3.3 Robust Estimator of Nonlinear Regression with Heteroscedastic Error Variances 192

9.3.4 The Performance of Robust Two Stage Estimator in Nonlinear Regression with Autocorrelated Error 192

9.3.5 Real Data Modelling 193 9.4 Recommendation for Further Research 194

REFERENCES 195 BIODATA OF STUDENT 204 LIST OF PUBLICATIONS 205

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